8 Cf. Or. Chald. 198 (with des Places's references); Syrianus, In met. 182, 24; Proclus, In Crat. 32, 22 and 28; Saffrey and Westernik's notes in Proclus, Theol. Plat. III 145; IV 120-1.

The 'Geminus' chapters in Proclus also contain arguments and theories which constitute elaborations on ideas first found in Iamblichus and (especially) in Syrianus. For example chapter 6 is devoted to arguing against the Aristotelian theory that numbers are mere abstractions from material objects. In the chapter corresponding to

end p.158

Proclus' preceding chapter (Comm. 34, 9-10) Iamblichus briefly rejects the theory of abstraction, but without argument. In his chapter 6 (12, 9 ff.) Proclus assembles arguments against Aristotle: how could mathematicals possess accuracy if they are derived from sensible objects? Where is the indivisibility in mathematicals to be found in sensibles where all is divisible? How could immutable laws be derived from ever-changing sensibles? How could the general be prior in demonstration if it is posterior to sensible particulars? Some of these arguments, expressed in the same interrogative and accumulative form, appear already in Syrianus. ⁹ 

⁹ Syrianus, In met. 95, 29-38; 90, 17-23; cf. above, p. 133. Proclus returns to the polemic against abstraction (applied to geometricals) in In Eucl. 49, 12 ff.; 139, 26-140, 18; In Parm. 894, 24 ff.; 980, 17 ff.

This suggests that Proclus, in chapter 6 of his prologue, is strengthening the anti-Aristotelian position of Iamblichus with the help of the anti-Aristotelian arguments that he had learnt from Syrianus. Proclus also rejects in chapter 6 the Aristotelian idea that the soul is a tabula rasa (16, 8-10). It is rather 'a tablet that has always been inscribed and is always writing itself and being written on by intellect.' Proclus is assuming a theory of innate ideas (here in particular mathematicals) that derive from intellect and are unfolded by the soul, a theory in which Aristotle's image of the soul as tablet (γ αμματει ον, De an. 430 a 1) is given a Platonic turn. ¹⁰ 

¹⁰ Cf. also In Eucl. 186, 6-7; In Crat. 26, 26-7; In Alc. §§ 277, 10-18; 280, 19-281, 7.

The origin of this Platonic interpretation of the image of the De anima appears to be Iamblichus. ¹¹ 

¹¹ Cf. Philoponus, In De an. 533, 25 ff.; Steel (1978), 148; Segonds (1985-6) II 435; cf. Plotinus V 3, 4, 20-3.

In the same chapter finally Proclus explains mathematicals as 'projections' (π οβολα ) by the soul of innate intelligible principles. ¹² 

¹² In Eucl. 17, 22-18, 4; cf. also 13, 6 ff.; 52, 20 ff.; 78, 20 ff.; 141, 2 ff.; below, pp. 200-1. On the theory of mathematicals as projections in Proclus, cf., for example, Breton (1969), 28-31, 111-22; Charles (1982), 191-201; Beierwaltes (1985), 258 ff.

This is again a theory first clearly found in Syrianus and which appears to have its ultimate origin in Iamblichus (above, pp. 133-4).

The place in the history of philosophy of the elaborate Platonic epistemology of the sixth chapter of Proclus' prologue is not then the first century ad . It represents rather a developed stage of the anti-Aristotelian arguments and Platonic theories first found in Syrianus and whose ultimate source of inspiration appears to be Iamblichus. We may conclude in other words that the author of this chapter is Proclus himself. The occasion that provoked it can be found in the

end p.159

chapter in Iamblichus corresponding to Proclus' chapter 5: a mention of the Aristotelian theory of abstraction. The technical use of Chaldaean terminology in chapter 2 also shows that this chapter cannot simply be dated to the first century ad , but indicates a date no earlier than Porphyry and very probably later.

The hypothesis that much of Proclus' prologue is not based on Iamblichus' On Pythagoreanism III, but is to be assigned to an earlier source from which Iamblichus also was inspired, then (1) involves a number of anachronisms; (2) makes the questionable assumption that Proclus is not himself the source of the greater clarity and order of his prologue; and (3) ignores the evidence of Syrianus' opinion of Iamblichus' book and his recommendation of it to students as a manual of general mathematics.

As the more obvious and likely thesis that Proclus is using Iamblichus has not been examined in any detail, it is desirable to compare the two authors from this viewpoint. Chapter 6 in Proclus comes after a chapter which corresponds to Iamblichus' text, but it does not itself have any equivalent in Iamblichus. The explanation of this has already been found: Proclus is elaborating in chapter 6, in the light of Syrianus' teaching, a defence of Iamblichus' rejection of Aristotelian abstraction in the chapter corresponding to Proclus' fifth chapter. Explanations can also be found from this point-of-view for other additions and omissions in Proclus' prologue as compared to Iamblichus' text. ¹³ 

¹³ For example, nothing in Proclus' prologue corresponds to Iamblichus, Comm. ch. 4: an explanation of this is suggested by a scholium on Iamblichus' chapter: 'Here the author introduces intelligible matter, as did the Pythagoreans before him, but Proclus does not agree with them' (Comm. 100, 11-13). At 8, 7 Proclus refers to 'they call . . . '; who 'they' are is not at all clear from Proclus' text; however, Iamblichus, in the corresponding chapter (ch. 5), is summarizing Pythagorean doctrine.

I would like to examine in particular the case of chapter 8, which has not been attributed to Geminus (although it has a corresponding section in Iamblichus, Comm. chs. 15-16), since this case will reveal not only Proclus' way of proceeding vis-à-vis Iamblichus' text, but also introduce themes of some importance to our larger interests. ¹⁴ 

¹⁴ Cf. Mueller's analysis (1987b) of this chapter; he notes many of the facts that I wish to mention.

In comparing chapter 8 of Proclus' prologue with chapters 15-16 of Iamblichus' On Pythagoreanism III, one notices first a difference in the way sources are cited. In Iamblichus, for instance, we find an undeclared allusion to Plotinus I 3, 3, 5-7 (Comm. 55, 16-19); in Proclus Plotinus is named and the Plotinian text is more carefully reproduced

end p.160

(21, 20-4). On the supposition then that Proclus is using Iamblichus, we must conclude that Proclus recognized the Plotinian allusion and replaced it with a more faithful quotation. This is not at all unlikely: Proclus, after all, wrote a Commentary on Plotinus, a singular honour he reserved otherwise for the ancient philosophers and theologians. ¹⁵ 

¹⁵ Cf. Westerink (1959). Of course Proclus' quotation of Plotinus is, by modern standards, loose.

The same phenomenon can be observed in relation to the two authors' use of Plato: Proclus quotes explicitly and more fully the texts of Plato implicitly used by Iamblichus ¹⁶ 

¹⁶ Compare Iambl. 55, 8-12, with Procl. 20, 14-26; Iambl. 57, 23-58, 4, with Procl. 25, 3-11.

and especially quotes texts from Plato which support and illustrate points made in Iamblichus. ¹⁷ 

¹⁷ Compare Iambl. 55, 22-56, 4, with Procl. 22, 17-23, 11; Iambl. 56, 8-13, with Procl. 24, 4-20.

If then Proclus is indeed using Iamblichus, he is taking advantage of his immense learning to revise the text by naming, quoting more fully, and supplementing Iamblichus' undeclared sources.

A closer comparative examination of the chapters in question in Proclus and Iamblichus strengthens the impression that Proclus is reworking, expanding, and clarifying the Iamblichean chapters. Both authors are concerned with showing the importance of mathematics for theology (metaphysics), physics, politics, ethics, and the productive arts. The Iamblichean text has been summarized above (pp. 48, 50); I shall therefore confine myself here to noting some of the differences in Proclus' version.

(a) Theology. After his undeclared allusion to Plotinus I 3, 3 Iamblichus concludes his section on this subject thus: 'For all such things provide a major starting point for knowledge of beings and intelligibles' (55, 21-2). Having named and quoted Plotinus more fully, Proclus says:

But that mathematics produces a contribution of the first order to philosophy is clear from these points. But it is necessary to recall the individual and show that for theology mathematics prepares intellectual insight. For those truths about the gods that are difficult for imperfect minds to discover and understand, these mathematical theories, through images, show to be trustworthy, evident, and irrefutable. ¹⁸